Video Transcript
In this video, we’ll learn how to
multiply monomials involving single and multiple variables. We’ll begin, of course, by
recalling what we actually mean by the term monomial.
The word mono- is from the ancient
Greeks. “Monos” meant alone, only, solo, or
single. And so, we say a monomial is a
polynomial with just one term, in other words, a single term that’s made up of
numbers and/or variables. They might have multiple variables,
but the exponents of any part of the term will only ever be positive integers. For instance, three 𝑥 to the
fourth power 𝑦 is a monomial. But two 𝑥 to the power of negative
five is not, since its exponent is negative. Similarly, a radical or a surd is
not considered to be a monomial. And this is because we can
represent radicals as fractional exponents. And of course, a fraction is not an
integer. So, we’re going to look at
multiplying two or more monomials. And we’re going to begin simply by
looking at a problem involving no algebraic terms.
Find the value of three times three
cubed times three squared.
We’re being asked to find the value
of this product. And so, what we could do is begin
by working out the value of three cubed and three squared. Three squared, of course, is the
same as three times three, which is equal to nine. And then, three cubed is three
times three times three, which is equal to 27. And so, to find the value of three
times three cubed times three squared, we’re going to work out three times 27 times
nine. Now, of course, multiplication is
commutative. It can be performed in any
order. So, let’s begin by working out
three times nine. Three multiplied by nine is 27. So, three times 27 times nine is
the same as 27 times 27. And of course, we can use any
method to calculate this.
Let’s recall how we might use the
column method. Seven times seven is 49. So, we put a nine here, and we
carry the four. Two times seven is 14, and then
when we add this four, we get 18. And so, we found that 27 times
seven is 189. We’re now going to multiply the
digits two and seven by this two. Now, of course, this two is in the
tens column, so it’s equivalent to multiplying by 20. And so, we can add a zero here to
take this into account. Seven times two is 14. So, we put a four here and carry
the one. Then, two times two is four, and
when we add this one, we get five. So, 27 times 20 is 540. We’re now going to add these two
numbers. Nine plus zero is nine; eight plus
four is 12. And so, we carry the one. Then, one plus five plus that
carried one is seven. And so, 27 times 27 is 729. And of course, we can therefore say
that that’s the value of our product.
But was there another way we
could’ve done this? Well, yes, we have a rule for
working with exponential terms. As long as the base is the same, to
multiply these sorts of terms, we simply add their exponents. The base in this general rule is
𝑥. And so, 𝑥 to the power of 𝑎 times
𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. And we can extend this rule into
multiplying three terms. If we think of three as three to
the first power, we can say that three to the first power times three cubed times
three squared is the same as three to the power of one plus three plus two, which is
three to the sixth power. Now, of course, we’ve really just
simplified our product. We do need to still evaluate it to
get 729. Either way, we find the value of
three times three cubed times three squared to be equal to 729.
And so, we’ve looked at how we can
simplify and evaluate products of monomials made up of purely numerical terms. So, what do we do when we work with
products of numbers and variables?
Remember this dot is used
interchangeably with the multiplication symbol. And so, to simplify the expression
given, we’re going to recall some of the properties of multiplication. Firstly, the associative property
of multiplication. This says that when three or more
numbers are multiplied, the product is the same, no matter how those numbers are
grouped. And so, we don’t really need the
parentheses here. It will be the same as just doing
six times 𝑥 times eight. Then, the commutative property says
that the product will be the same even if we switch the numbers around. For instance, three times four is
the same as four times three. And so, we’re going to switch the
𝑥 and the eight around and rewrite our problem as six times eight times 𝑥. And then, of course, since the
grouping doesn’t really matter, we can evaluate six times eight. Six times eight is 48, so this
becomes 48 times 𝑥. And of course, we know that we can
simply write that as 48𝑥. So, when we simplify the expression
six times eight times 𝑥, we get 48𝑥.
And so, we’ve seen now how to
multiply fairly simple monomials using some of the multiplication properties, along
with the laws for working with exponents. We’re now going to combine these
and look at some more complicated problems.
Simplify seven 𝑥 to the fourth
power times eight 𝑥 to the seventh power.
There are two things we know about
multiplication. It is associative and
commutative. The associative property tells us
that the product will be the same no matter how the numbers are grouped. And the commutative property tells
us that the order in which we perform the calculation really doesn’t matter. And so, we’re going to begin by
sort of unsimplifying each term in our expression. When we do, we get seven times 𝑥
to the fourth power times eight times 𝑥 to the seventh power. We’re now going to switch around 𝑥
to the fourth power and eight. And so, we see that our expression
now becomes seven times eight times 𝑥 to the fourth power times 𝑥 to the seventh
power. And in fact, we can work out the
numerical part. We know that seven times eight is
56, which means that this becomes 56 times 𝑥 to the fourth power times 𝑥 to the
seventh power.
But what do we do with these
algebraic parts? Well, we have a rule for
multiplying exponential terms. As long as the base is the same, we
add the exponents. So, 𝑥 to the power of 𝑎 times 𝑥
to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. This in turn means that 𝑥 to the
fourth power times 𝑥 to the seventh power is 𝑥 to the power of four plus
seven. And since four plus seven is 11,
this becomes 𝑥 to the 11th power. Now, in fact, we know that we don’t
really want to include that multiplication symbol. And so, we simplify this further to
get 56𝑥 to the 11th power. And so, when we simplify seven 𝑥
to the fourth power times eight 𝑥 to the seventh power, we get 56𝑥 to the 11th
power.
Now, actually, all this sort of
unsimplifying and then switching terms around is quite long winded, so we can
actually generalize. We say that to multiply two or more
monomials, we can first multiply the coefficients, those are the numbers in front of
the algebraic parts, and then separately multiply the variables, of course, using
the laws of exponents when necessary. Let’s apply this to a context-based
question to begin with.
Find an expression for the volume
of the rectangular prism shown.
And then, we have a rectangular
prism or a cuboid where we’ve been given the three dimensions. Let’s call this dimension here the
length, which I’ve abbreviated to 𝑙. We’ll call this dimension here 𝑤
for width, and then we’ll say this dimension is the height ℎ. And then, we know, of course, that
the volume of a rectangular prism is just the product of these three dimensions. It’s length times width times
height, meaning that the volume of our rectangular prism in cubic units will be
three 𝑥 times 10𝑥 times 15𝑥. And so, we’re actually finding the
product of three monomials. And we recall that to do this, we
first multiply the coefficients; those are the numerical parts. So, we’re going to begin by doing
three times 10 times 15. And then, we separately multiply
the variables. So here, we’re going to do 𝑥 times
𝑥 times 𝑥.
We do also know that multiplication
is commutative, so it can be done in any order. So, we’ll begin by multiplying the
three by the 15 to get 45. And then 45 multiplied by this 10
here is 450. And so, when we multiply the
coefficients of our monomials, we get 450. And now, we multiply the algebraic
parts. 𝑥 times 𝑥 times 𝑥 is 𝑥
cubed. And so, we can say that the volume
of the rectangular prism shown in cubic units is 450𝑥 cubed.
Let’s now consider what happens
when our monomials contain more than one variable.
Simplify three 𝑥 times four
𝑦.
Here, we’re looking to find the
product of two monomials. The first is three 𝑥 and the
second is four 𝑦. And so, we can recall that when we
multiply monomials, we begin by multiplying the coefficients. That’s the numbers in front of the
algebraic parts. And then, we can multiply the
variables using our laws for exponents, if necessary. Now, the coefficients of our two
monomials are three and four. So, we first work out three times
four, which is equal to 12. Then, we see, of course, that the
variable parts are 𝑥 and 𝑦. So, we do 𝑥 times 𝑦, which we
just write as 𝑥𝑦. Three 𝑥 times four 𝑦 then will
just be the product of these terms. It’s 12 times 𝑥𝑦 that we can
write as 12𝑥𝑦. And so, we’ve simplified three 𝑥
times four 𝑦, and we got 12𝑥𝑦.
Let’s now combine everything we’ve
done into looking at one slightly nastier example.
Simplify six 𝑥 squared 𝑦 squared
times negative nine 𝑥 squared 𝑦 to the fifth power.
Here, we’re finding the product of
two monomials. And so, we recall the process for
multiplying two or more monomials. We begin by multiplying the
coefficients of each term. And then we separately multiply the
variables. Now, we will need to use the laws
of exponents to do so. The specific law that we’re going
to use is the fact that when the bases are the same, we’d multiply exponential terms
by just adding the exponents. So, 𝑥 to the power of 𝑎 times 𝑥
to the power of 𝑏 is 𝑥 to the power of 𝑎 plus 𝑏. Let’s identify the coefficients in
our product. We have six here and negative nine
here, and so we’re going to multiply six by negative nine. Six times nine is 54. And a positive multiplied by a
negative is a negative, so six times negative nine is negative 54.
Next, we multiply the
variables. We’re going to multiply 𝑥 squared
by 𝑥 squared. And of course, to do so, we simply
add the exponents. So, we get 𝑥 to the power of two
plus two, which is, of course, the same as 𝑥 to the fourth power. There’s another variable we need to
consider though, and that’s 𝑦. So, we do the same with this
one. 𝑦 squared times 𝑦 to the fifth
power is the same as 𝑦 to the power of two plus five. And of course, since two plus five
is seven, 𝑦 squared times 𝑦 to the fifth power is 𝑦 to the seventh power. When we combine all of this, we get
the result of multiplying six 𝑥 squared 𝑦 squared by negative nine 𝑥 squared 𝑦
to the fifth power. It’s negative 54𝑥 to the fourth
power 𝑦 to the seventh power.
We’re now going to recap the key
points from this lesson. In this video, we learned that a
monomial is a polynomial with just one term. We know that that single term can
be made up of numbers and/or variables, and it might even contain more than one
variable, but that any exponents in that expression would always be positive
integers. We would never include, say, a
fractional exponent nor a negative one. We also saw that to multiply two or
more monomials, we multiply the coefficients and then we separately multiply the
variables using the laws of exponents where necessary. And the law for exponents that we
used was 𝑥 to the power of 𝑎 times 𝑥 to the power of 𝑏 is 𝑥 to the power of 𝑎
plus 𝑏. In other words, as long as the base
is the same, we simply add the exponents.
